$k$Forrelation Optimally Separates Quantum and Classical Query Complexity
Abstract
Aaronson and Ambainis (SICOMP `18) showed that any partial function on $N$ bits that can be computed with an advantage $\delta$ over a random guess by making $q$ quantum queries, can also be computed classically with an advantage $\delta/2$ by a randomized decision tree making ${O}_q(N^{1\frac{1}{2q}}\delta^{2})$ queries. Moreover, they conjectured the $k$Forrelation problem  a partial function that can be computed with $q = \lceil k/2 \rceil$ quantum queries  to be a suitable candidate for exhibiting such an extremal separation. We prove their conjecture by showing a tight lower bound of $\widetilde{\Omega}(N^{11/k})$ for the randomized query complexity of $k$Forrelation, where the advantage $\delta = 2^{O(k)}$. By standard amplification arguments, this gives an explicit partial function that exhibits an $O_\epsilon(1)$ vs $\Omega(N^{1\epsilon})$ separation between boundederror quantum and randomized query complexities, where $\epsilon>0$ can be made arbitrarily small. Our proof also gives the same bound for the closely related but nonexplicit $k$Rorrelation function introduced by Tal (FOCS `20). Our techniques rely on classical Gaussian tools, in particular, Gaussian interpolation and Gaussian integration by parts, and in fact, give a more general statement. We show that to prove lower bounds for $k$Forrelation against a family of functions, it suffices to bound the $\ell_1$weight of the Fourier coefficients between levels $k$ and $(k1)k$. We also prove new interpolation and integration by parts identities that might be of independent interest in the context of rounding highdimensional Gaussian vectors.
 Publication:

arXiv eprints
 Pub Date:
 August 2020
 arXiv:
 arXiv:2008.07003
 Bibcode:
 2020arXiv200807003B
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity
 EPrint:
 40 pages, 2 figures. Change from v1 to v2: Updated figures to fix an Adobe Acrobat specific issue. Change from v0 to v1: Improved the advantage $\delta$ to $2^{O(k)}$ strengthening the main conclusions. Added a reference to the independent work of Sherstov, Storozhenko and Wu (arxiv:2008.10223) who obtained a similar lower bound for the randomized query complexity of $k$Rorrelation